Normalized defining polynomial
\( x^{30} - x^{29} - 63 x^{28} + 76 x^{27} + 1638 x^{26} - 2211 x^{25} - 23247 x^{24} + 33685 x^{23} + 200912 x^{22} - 305788 x^{21} - 1113890 x^{20} + 1764076 x^{19} + 4052792 x^{18} - 6681693 x^{17} - 9698303 x^{16} + 16831893 x^{15} + 14983577 x^{14} - 28119106 x^{13} - 14238718 x^{12} + 30607456 x^{11} + 7427701 x^{10} - 21042174 x^{9} - 1386542 x^{8} + 8779409 x^{7} - 421848 x^{6} - 2122774 x^{5} + 259157 x^{4} + 271149 x^{3} - 46850 x^{2} - 14052 x + 2969 \)
Invariants
| Degree: | $30$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[30, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(23984756244761087654296442851217575634882843017578125=5^{15}\cdot 7^{20}\cdot 11^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.72$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $5, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(385=5\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{385}(256,·)$, $\chi_{385}(1,·)$, $\chi_{385}(4,·)$, $\chi_{385}(214,·)$, $\chi_{385}(71,·)$, $\chi_{385}(64,·)$, $\chi_{385}(9,·)$, $\chi_{385}(331,·)$, $\chi_{385}(141,·)$, $\chi_{385}(16,·)$, $\chi_{385}(81,·)$, $\chi_{385}(86,·)$, $\chi_{385}(344,·)$, $\chi_{385}(324,·)$, $\chi_{385}(284,·)$, $\chi_{385}(221,·)$, $\chi_{385}(289,·)$, $\chi_{385}(291,·)$, $\chi_{385}(36,·)$, $\chi_{385}(169,·)$, $\chi_{385}(144,·)$, $\chi_{385}(366,·)$, $\chi_{385}(114,·)$, $\chi_{385}(179,·)$, $\chi_{385}(309,·)$, $\chi_{385}(246,·)$, $\chi_{385}(361,·)$, $\chi_{385}(379,·)$, $\chi_{385}(254,·)$, $\chi_{385}(191,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{881} a^{28} - \frac{333}{881} a^{27} - \frac{163}{881} a^{26} + \frac{193}{881} a^{25} + \frac{328}{881} a^{24} - \frac{388}{881} a^{23} + \frac{119}{881} a^{22} + \frac{218}{881} a^{21} + \frac{153}{881} a^{20} - \frac{126}{881} a^{19} - \frac{72}{881} a^{18} + \frac{381}{881} a^{17} + \frac{32}{881} a^{16} + \frac{78}{881} a^{15} + \frac{26}{881} a^{14} - \frac{326}{881} a^{13} - \frac{312}{881} a^{12} - \frac{173}{881} a^{11} + \frac{219}{881} a^{10} + \frac{408}{881} a^{9} + \frac{211}{881} a^{8} + \frac{136}{881} a^{7} - \frac{223}{881} a^{6} + \frac{302}{881} a^{5} - \frac{201}{881} a^{4} + \frac{192}{881} a^{3} - \frac{39}{881} a^{2} - \frac{222}{881} a - \frac{11}{881}$, $\frac{1}{57851956364409667752394883429121997797351297535215975690295747109} a^{29} + \frac{21172931878604696161787311595512536395024528798318776913731145}{57851956364409667752394883429121997797351297535215975690295747109} a^{28} - \frac{16699802327725531396686134556026410153113747576808954079355997715}{57851956364409667752394883429121997797351297535215975690295747109} a^{27} + \frac{25537319886136116735191647695650313584632668968283408610969454266}{57851956364409667752394883429121997797351297535215975690295747109} a^{26} - \frac{605275450156711300359940075721790367532855675257603788112640801}{57851956364409667752394883429121997797351297535215975690295747109} a^{25} - \frac{5974748798366158270830708466975176565604003149784123307103228969}{57851956364409667752394883429121997797351297535215975690295747109} a^{24} + \frac{6384686229946910423644224874978345515413672269741891584776376012}{57851956364409667752394883429121997797351297535215975690295747109} a^{23} + \frac{27539695108742680552010181594360456370930110387677258240300399404}{57851956364409667752394883429121997797351297535215975690295747109} a^{22} - \frac{11174112590834331188580179280722267525887111911592663009729146427}{57851956364409667752394883429121997797351297535215975690295747109} a^{21} - \frac{19457477903261648752849962802178797978056109079916891973134671027}{57851956364409667752394883429121997797351297535215975690295747109} a^{20} + \frac{24350019074993252407740316316291644873223674884123636143715771802}{57851956364409667752394883429121997797351297535215975690295747109} a^{19} + \frac{21157844313622608168817405009183506582302108740238655257729108815}{57851956364409667752394883429121997797351297535215975690295747109} a^{18} + \frac{7268444714950398143533514518432863512619086816954567994010572197}{57851956364409667752394883429121997797351297535215975690295747109} a^{17} - \frac{10791350886684131335719918443719078794015639986099922109884645305}{57851956364409667752394883429121997797351297535215975690295747109} a^{16} + \frac{18006553818951770270626373916056656408942974206171040631815841514}{57851956364409667752394883429121997797351297535215975690295747109} a^{15} + \frac{12983894700796361794699734401775953733809269760883377034262092114}{57851956364409667752394883429121997797351297535215975690295747109} a^{14} + \frac{6671040617986217179635270885020136018221232104347701824625648257}{57851956364409667752394883429121997797351297535215975690295747109} a^{13} + \frac{20769340556407796184727633727650362577739241797470775928238153372}{57851956364409667752394883429121997797351297535215975690295747109} a^{12} + \frac{6406042113027382233157266991057653346536123862721419168866375207}{57851956364409667752394883429121997797351297535215975690295747109} a^{11} - \frac{23865066064634712452735954980461897393889320216823795941859264393}{57851956364409667752394883429121997797351297535215975690295747109} a^{10} + \frac{25839401133935444765243179860624976412037189544852234782258110884}{57851956364409667752394883429121997797351297535215975690295747109} a^{9} - \frac{28829489038112001335400974695147784030977451952361590043527450956}{57851956364409667752394883429121997797351297535215975690295747109} a^{8} + \frac{3811640935633469614285608246489887300655283477632328922990803739}{57851956364409667752394883429121997797351297535215975690295747109} a^{7} - \frac{2030020618653833607318525624722532078806710758117101201816980847}{57851956364409667752394883429121997797351297535215975690295747109} a^{6} + \frac{26583405854049692875305458568230430418199630933140783725394589465}{57851956364409667752394883429121997797351297535215975690295747109} a^{5} - \frac{22636721923522010961918457630643388675105606216285138121250751336}{57851956364409667752394883429121997797351297535215975690295747109} a^{4} - \frac{17323939694850452349840119656375755349294732739919406558654223426}{57851956364409667752394883429121997797351297535215975690295747109} a^{3} - \frac{176602195479981620557599088030474808214187978342555154370415574}{57851956364409667752394883429121997797351297535215975690295747109} a^{2} - \frac{11629403079731949128816279349647529486635112454484513649269429426}{57851956364409667752394883429121997797351297535215975690295747109} a + \frac{22036863499070834028978108861912988753225971574667577675153197538}{57851956364409667752394883429121997797351297535215975690295747109}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $29$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 29415349795758660 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.6.300125.1, 10.10.669871503125.1, 15.15.886528337182930278529.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $30$ | $30$ | R | R | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ | $30$ | $15^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ | $30$ | $30$ | $15^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| $11$ | 11.15.12.1 | $x^{15} + 165 x^{10} + 5324 x^{5} + 323433$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |
| 11.15.12.1 | $x^{15} + 165 x^{10} + 5324 x^{5} + 323433$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ | |